intr_elim.ML

INSTALLATION OF INDUCTIVE DEFINITIONS
HOL/Sexp, List, LList: updated for inductive defs; streamlined proofs
HOL/List, Subst/UTerm, ex/Simult, ex/Term: updated refs to Sexp intr rules
HOL/Univ/diag_eqI: new
HOL/intr_elim: now checks that the inductive name does not clash with
existing theory names
HOL/Sum: now type + is an infixr, to agree with type *
HOL/Set: added Pow and the derived rules PowI, PowD, Pow_bottom, Pow_top
HOL/Fun/set_cs: now includes Pow rules
HOL/mono/Pow_mono: new
HOL/Makefile: now has Inductive.thy,.ML and ex/Acc.thy,.ML

HOL/Sexp,List,LList,ex/Term: converted as follows
node *set -> item
Sexp -> sexp
LList_corec -> <self>
LList_ -> llist_
LList\> -> llist
List_case -> <self>
List_rec -> <self>
List_ -> list_
List\> -> list
Term_rec -> <self>
Term_ -> term_
Term\> -> term

(*  Title: 	HOL/intr_elim.ML
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Introduction/elimination rule module -- for Inductive/Coinductive Definitions
*)

signature INDUCTIVE_ARG =	(** Description of a (co)inductive def **)
  sig
  val thy        : theory               (*new theory with inductive defs*)
  val monos      : thm list		(*monotonicity of each M operator*)
  val con_defs   : thm list		(*definitions of the constructors*)
  end;

(*internal items*)
signature INDUCTIVE_I =
  sig
  val rec_tms    : term list		(*the recursive sets*)
  val intr_tms   : term list		(*terms for the introduction rules*)
  end;

signature INTR_ELIM =
  sig
  val thy        : theory               (*copy of input theory*)
  val defs	 : thm list		(*definitions made in thy*)
  val mono	 : thm			(*monotonicity for the lfp definition*)
  val unfold     : thm			(*fixed-point equation*)
  val intrs      : thm list		(*introduction rules*)
  val elim       : thm			(*case analysis theorem*)
  val raw_induct : thm			(*raw induction rule from Fp.induct*)
  val mk_cases : thm list -> string -> thm	(*generates case theorems*)
  val rec_names  : string list		(*names of recursive sets*)
  end;

(*prove intr/elim rules for a fixedpoint definition*)
functor Intr_elim_Fun
    (structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end  
     and Fp: FP) : INTR_ELIM =
struct
open Logic Inductive Ind_Syntax;

val rec_names = map (#1 o dest_Const o head_of) rec_tms;
val big_rec_name = space_implode "_" rec_names;

val _ = deny (big_rec_name  mem  map ! (stamps_of_thy thy))
             ("Definition " ^ big_rec_name ^ 
	      " would clash with the theory of the same name!");

(*fetch fp definitions from the theory*)
val big_rec_def::part_rec_defs = 
  map (get_def thy)
      (case rec_names of [_] => rec_names | _ => big_rec_name::rec_names);


val sign = sign_of thy;

(********)
val _ = writeln "  Proving monotonicity...";

val Const("==",_) $ _ $ (Const(_,fpT) $ fp_abs) =
    big_rec_def |> rep_thm |> #prop |> unvarify;

(*For the type of the argument of mono*)
val [monoT] = binder_types fpT;

val mono = 
    prove_goalw_cterm [] 
      (cterm_of sign (mk_tprop (Const("mono", monoT-->boolT) $ fp_abs)))
      (fn _ =>
       [rtac monoI 1,
	REPEAT (ares_tac (basic_monos @ monos) 1)]);

val unfold = standard (mono RS (big_rec_def RS Fp.Tarski));

(********)
val _ = writeln "  Proving the introduction rules...";

fun intro_tacsf disjIn prems = 
  [(*insert prems and underlying sets*)
   cut_facts_tac prems 1,
   rtac (unfold RS ssubst) 1,
   REPEAT (resolve_tac [Part_eqI,CollectI] 1),
   (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   rtac disjIn 1,
   REPEAT (ares_tac [refl,exI,conjI] 1)];

(*combines disjI1 and disjI2 to access the corresponding nested disjunct...*)
val mk_disj_rls = 
    let fun f rl = rl RS disjI1
	and g rl = rl RS disjI2
    in  accesses_bal(f, g, asm_rl)  end;

val intrs = map (uncurry (prove_goalw_cterm part_rec_defs))
            (map (cterm_of sign) intr_tms ~~ 
	     map intro_tacsf (mk_disj_rls(length intr_tms)));

(********)
val _ = writeln "  Proving the elimination rule...";

(*Includes rules for Suc and Pair since they are common constructions*)
val elim_rls = [asm_rl, FalseE, Suc_neq_Zero, Zero_neq_Suc,
		make_elim Suc_inject, 
		refl_thin, conjE, exE, disjE];

(*Breaks down logical connectives in the monotonic function*)
val basic_elim_tac =
    REPEAT (SOMEGOAL (eresolve_tac (elim_rls@sumprod_free_SEs)
	      ORELSE' bound_hyp_subst_tac))
    THEN prune_params_tac;

val elim = rule_by_tactic basic_elim_tac (unfold RS equals_CollectD);

(*Applies freeness of the given constructors, which *must* be unfolded by
  the given defs.  Cannot simply use the local con_defs because con_defs=[] 
  for inference systems. *)
fun con_elim_tac defs =
    rewrite_goals_tac defs THEN basic_elim_tac THEN fold_tac defs;

(*String s should have the form t:Si where Si is an inductive set*)
fun mk_cases defs s = 
    rule_by_tactic (con_elim_tac defs)
      (assume_read thy s  RS  elim);

val defs = big_rec_def::part_rec_defs;

val raw_induct = standard ([big_rec_def, mono] MRS Fp.induct);
end;